Research Article FixedPointTheoryandtheLiouville–CaputoIntegro-Differential FBVPwithMultipleNonlinearTerms ShahramRezapour , 1,2 AliBoulfoul, 3 BrahimTellab , 3 MohammadEsmaelSamei , 4 SinaEtemad , 1 andRenyGeorge 5,6 1 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran 2 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan 3 Laboratory of Applied Mathematics, Kasdi Merbah University, Ouargla B.P. 511 30000, Algeria 4 Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan, Iran 5 Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia 6 Department of Mathematics and Computer Science, St. omas College, Bhilai, Chhattisgarth 49006, India Correspondence should be addressed to Reny George; renygeorge02@yahoo.com Received 19 October 2021; Revised 6 January 2022; Accepted 17 January 2022; Published 24 February 2022 Academic Editor: Alexander Meskhi Copyright © 2022 Shahram Rezapour et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is work is reserved for the study of a special category of boundary value problems (BVPs) consisting of Liouville–Caputo integro-differential equations with multiple nonlinear terms. is fractional model and its boundary value conditions (BVCs) involve different simple BVPs, in which the second BVC as a linear combination of two Caputo derivatives of the unknown function equals a nonzero constant. e Banach principle gives a unique solution for this Liouville–Caputo BVP. Further, the Krasnoselskii and Leray–Schauder criteria give the existence property regarding solutions of the mentioned problem. For each theorem, we provide an example based on the required hypotheses and derive numerical data in the framework of tables and figures to show the consistency of results from different points of view. 1.Introduction In recent years, fractional differential equations have attracted the attention of many authors because of the numerous applications in various branches of science and engineering, in particular, fluid mechanics, image and signal processing, electromagnetic theory, potential the- ory, fractals theory, biology, control theory, viscoelas- ticity, and so on [1–3]. From the mathematical point of view, a number of researchers working on fractional calculus conduct their research in the field of applications of different fractional operators and various structures of BVPs in modeling abstract and real-world phenomena, but the discussion related to the fractional derivatives is an old problem and continues to receive many kinds of feedback. e physical aspect of the fractional derivative is now proved in many investigations. As we know, frac- tional-order derivatives have many advantages in com- parison to the first-order derivatives. For example, one of the most simple examples in which the fractional de- rivative has a significant impact can be observed in dif- fusion processes. It is established that the subdiffusion is obtained when the order of the fractional derivative be- longs to the interval (0, 1). Another impact of fractional derivatives can be observed in stability analysis. ere are many differential equations that are not stable with the first-order derivative but are stable when we replace the first-order derivative by the fractional-order derivatives. By considering these cases, we can understand the im- portance of fractional operators, and the Liou- ville–Caputo derivative is one of the most important examples in this field. For better and more accurate Hindawi Journal of Function Spaces Volume 2022, Article ID 6713533, 18 pages https://doi.org/10.1155/2022/6713533